Simplify and expand the following expression: $ \dfrac{3k}{5k + 4}+\dfrac{3k}{2k - 10} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5k + 4)(2k - 10)$ Multiply the first term by $\dfrac{2k - 10}{2k - 10}$ $ \begin{align*} \dfrac{3k}{5k + 4} \times \dfrac{2k - 10}{2k - 10} & = \dfrac{(3k)(2k - 10)}{(5k + 4)(2k - 10)} \\ & = \dfrac{6k^2 - 30k}{(5k + 4)(2k - 10)}\end{align*} $ Multiply the second term by $\dfrac{5k + 4}{5k + 4}$ $ \begin{align*} \dfrac{3k}{2k - 10} \times \dfrac{5k + 4}{5k + 4} & = \dfrac{(3k)(5k + 4)}{(2k - 10)(5k + 4)} \\ & = \dfrac{15k^2 + 12k}{(2k - 10)(5k + 4)}\end{align*} $ Now we have: $ = \dfrac{6k^2 - 30k}{(5k + 4)(2k - 10)} + \dfrac{15k^2 + 12k}{(2k - 10)(5k + 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{6k^2 - 30k + 15k^2 + 12k}{(5k + 4)(2k - 10)} $ $ = \dfrac{21k^2 - 18k}{(5k + 4)(2k - 10)}$ Expand the denominator: $ = \dfrac{21k^2 - 18k}{10k^2 - 42k - 40}$